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User blog:Rgetar/Generalizations of Ordinal array functions and FGH
For Ordinal array functions see User blog:Rgetar/Definitions update. Here are my Ordinal array functions: Xa, where a is ordinal, X is array of ordinals, and Xa is ordinal. Then I started to designate arrays of ordinals as "larger" ordinals, for example, array of finite and countable ordinals 1, ω + 1, 10, ε0, 5 as uncountable ordinal Ω4 + Ω3(ω + 1) + Ω210 + Ωε0 + 5 etc. cp(X), leo(X), lest(X; α), X0 Here are new function cp(X) and new definitions of functions leo(X), lest(X; α), X0 cp means "cardinality part" leo means "last element of" lest means "last element set to" ("Element", since I named leo and lest when I considered X as array, now I consider X as "larger" ordinal, but I didn't change names of these functions). cp(X), leo(X) Ordinal X of cardinality card(X) can be represented as sum X = cp(X) + leo(X) such as cp(X) ≥ card(X), leo(X) < card(X), and X cannot be represented as X = α + β such as 0 < β < card(X). Examples. For finite X cp(X) = X, leo(X) = 0. For countable X in my blog Ordinal arithmetic I designated cp(X) as lp(X), that is "limit part", and leo(X) as fp(X), that is "finite part". For example, X = ω4 + ω32 + ω23 + ω5 + 15 cp(X) = ω4 + ω32 + ω23 + ω5 leo(X) = 15 Example for card(X) = Ω: X = Ω2 + Ω + ω + 1 cp(X) = Ω2 + Ω leo(X) = ω + 1 Generally, ordinal X of cardinality Ωβ can be represented as sum of terms Ωβiαi, where card(i) ≤ Ωβ, card(αi) < Ωβ, i are decreasing. leo(X) = Ωβ0α0 = α0 cp(X) is rest of this sum without α0. lest(X; α) lest(X; α), where card(α) < card(X), is X with leo(X) replaced with α. That is cp(lest(X; α)) = cp(X) leo(lest(X; α)) = α X0 X0 = lest(X; 0) That is cp(X0) = cp(X) leo(X0) = 0 Generalization of Ordinal array function Now Ordinal array function has subscript β: Xβα where α < Ωβ, X < Ωβ + 1, Xβα < Ωβ. Definition: 0βα = α + 1 + 1βα = X0βXβα Xβα = sup([Xn]βα), 1 < cof(X) < Ωβ Xβα = [Xα]βα, cof(X) = Ωβ where Xn is n-th element of fundamental sequence of X. So, Ordinal array function becomes dependent on fundamental sequence system (fss). β = 0 Special case of Xβα for β = 0: X0n where X is countable or finite ordinal, n is natural number, X0n is also natural number. Examples for Wainer hierarchy: 00n = n + 1 10n = n + 2 20n = n + 3 30n = n + 4 40n = n + 5 Generally, for natural m m0n = n + m + 1 ω0n = n0n = 2n + 1 + 10n = ω0ω0n = ω0(2n + 1) = 4n + 3 + 20n = ω0+ 10n = ω0(4n + 3) = 8n + 7 + 30n = ω0+ 20n = ω0(8n + 7) = 16n + 15 Generally, for natural m + m0n = 2m + 1(n + 1) - 1 ω20n = + n0n = 2n + 1(n + 1) - 1 + 10n = ω20ω20n + 20n = ω20+ 10n + 30n = ω20+ 20n ω30n = + n0n + 10n = ω30ω30n + 20n = ω30+ 10n + 30n = ω30+ 20n ω40n = + n0n ω50n = + n0n ω20n = ωn0n + 10n = ω20ω20n + 20n = ω20+ 10n + ω0n = + n0n + ω + 10n = + ω0+ ω0n + ω20n = + ω + n0n ω220n = + ωn0n ω230n = + ωn0n ω30n = ω2n0n ω40n = ω3n0n ω50n = ω4n0n ωω0n = ωn0n + 10n = ωω0ωω0n + 20n = ωω0+ 10n + ω0n = + n0n + ω20n = + ω + n0n + ω20n = + ωn0n ωω20n = + ωn0n + 10n = ωωn0n ωω20n = + n0n ωω20n = ωωn0n ωωω0n = ωωn0n Generalization of FGH X0n is similar to FGH (theirs inputs and outputs are natural numbers, their parameter is finite or countable ordinal, and they are both fss-dependent). And, as we have Ordinal array functions for β > 0, maybe, we can generalize FGH the same way? Yes, we can. Here is FGH: f0(n) = n + 1 fα + 1(n) = fαn(n) fα(n) = fαn(n), iff α is limit ordinal where n is natural number, α is countable ordinal or natural number, αn is n-th element of fundamental sequence of α, fαm + 1(n) = fα(fαm(n)), where m is natural number. fα(n) is also natural number. Generalization of FGH has two subscripts: fβ, X(α) where α < Ωβ, β is any ordinal, X < Ωβ + 1, fβ, X(α) < Ωβ. Definition: fβ, 0(α) = α + 1 fβ, X + 1(α) = fβ, Xα(α) fβ, X(α) = sup(fβ, Xn(α)), 1 < cof(X) < Ωβ fβ, X(α) = fβ, Xα(α), cof(X) = Ωβ where Xn is n-th element of fundamental sequence of X. And now may be infinite iteration: fβ, Xγ + 1(α) = fβ, X(fβ, Xγ(α)) fβ, Xγ(α) = sup(fβ, Xγn(α)), iff γ is limit ordinal. Special case of fβ, X(α) for β = 0: f0, X(α) = fX(α) — it is FGH. Category:Blog posts